ADDITIVE ρ-FUNCTIONAL EQUATIONS IN BANACH SPACES

Abstract

In this paper, we solve the additive ρ-functional equations

1. INTRODUCTION

The stability problem of functional equations originated from a question of Ulam [5] concerning the stability of group homomorphisms.

The functional equation is called the Cauchy equation. In particular, every solution of the Cauchy equation is said to be an additive mapping. Hyers [3] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [1] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by Gǎvruta [2] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach.

In Section 2, we solve the additive functional equation (0.1) and prove the Hyers-Ulam stability of the additive functional equation (0.1) in Banach spaces.

In Section 3, we solve the additive functional equation (0.2) and prove the Hyers-Ulam stability of the additive functional equation (0.2) in Banach spaces.

Throughout this paper, assume that X is a normed space and that Y is a Banach space.

 

2. ADDITIVE ρ-FUNCTIONAL EQUATION (0.1)

Let ρ be a number with ρ ≠ 1, 2.

We solve and investigate the additive ρ-functional equation (0.1) in normed spaces.

Lemma 2.1. If a mapping f : X → Y satisfies

for all x, y, z ∈ X, then f : X → Y is additive.

Proof. Assume that f : X → Y satisfies (2.1).

Letting x = y = z = 0 in (2.1), we get −2f(0) = −ρf(0). So f(0) = 0.

Letting y = x and z = 0 in (2.1), we get f(2x)−2f(x) = 0 and so f(2x) = 2f(x) for all x ∈ X. Thus

for all x ∈ X.

It follows from (2.1) and (2.2) that

and so f(x + y + z) = f(x) + f(y) + f(z) for all x, y, z ∈ X. Since f(0) = 0,

for all x, y ∈ X.   ☐

We prove the Hyers-Ulam stability of the additive ρ-functional equation (2.1) in Banach spaces.

Theorem 2.2. Let φ : X3 → [0, ∞) be a function and let f : X → Y be a mapping satisfying f(0) = 0 and

for all x, y, z ∈ X. Then there exists a unique additive mapping A : X → Y such that

for all x ∈ X.

Proof. Letting y = x and z = 0 in (2.4), we get

for all x ∈ X. So

for all x ∈ X. Hence

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.7) that the sequence is Cauchy for all x ∈ X. Since Y is a Banach space, the sequence converges. So one can define the mapping A : X → Y by

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.7), we get (2.5).

Now, let T : X → Y be another additive mapping satisfying (2.5). Then we have

which tends to zero as q → ∞ for all x ∈ X. So we can conclude that A(x) = T(x) for all x ∈ X. This proves the uniqueness of A.

It follows from (2.3) and (2.4) that

for all x, y, z ∈ X. So

for all x, y, z ∈ X. By Lemma 2.1, the mapping A : X → Y is additive.   ☐

Corollary 2.3. Let r > 1 and θ be nonnegative real numbers, and let f : X → Y be a mapping satisfying f(0) = 0 and

for all x, y, z ∈ X. Then there exists a unique additive mapping A : X → Y such that

for all x ∈ X.

Proof. Letting φ(x, y, z) := in Theorem 2.2, we get the desired result.   ☐

Theorem 2.4. Let φ : X3 → [0, ∞) be a function and let f : X → Y be a mapping satisfying f(0) = 0, (2.4) and

for all x, y, z ∈ X. Then there exists a unique additive mapping A : X → Y such that

for all x ∈ X.

Proof. It follows from (2.6) that

for all x ∈ X. Hence

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.10) that the sequence is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence converges. So one can define the mapping A : X → Y by

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.10), we get (2.9).

The rest of the proof is similar to the proof of Theorem 2.2.   ☐

Corollary 2.5. Let r < 1 and θ be nonnegative real numbers, and let f : X → Y be a mapping satisfying f(0) = 0 and (2.8). Then there exists a unique additive mapping A : X → Y such that

for all x ∈ X.

Proof. Letting φ(x, y, z) := in Theorem 2.4, we get the desired result.   ☐

 

3. ADDITIVE ρ-FUNCTIONAL EQUATION (0.2)

Let ρ be a number with ρ ≠ 1.

We solve and investigate the additive ρ-functional equation (0.2) in normed spaces.

Lemma 3.1. If a mapping f : X → Y satisfies

for all x, y, z ∈ X, then f : X → Y is additive.

Proof. Assume that f : X → Y satisfies (3.1).

Letting x = y = z = 0 in (2.1), we get -2f(0) = -2ρf(0). So f(0) = 0.

Letting y = x and z = 0 in (2.1), we get f(2x)-2f(x) = 0 and so f(2x) = 2f(x) for all x ∈ X. Thus

for all x ∈ X.

It follows from (3.1) and (3.2) that

and so for all x, y ∈ X.   ☐

We prove the Hyers-Ulam stability of the additive ρ-functional equation (3.1) in Banach spaces.

Theorem 3.2. Let φ : X3 → [0, ∞) be a function and let f : X → Y be a mapping satisfying f(0) = 0 and

for all x, y, z ∈ X. Then there exists a unique additive mapping A : X → Y such that

for all x ∈ X.

Proof. Letting y = x and z = 0 in (3.3), we get

for all x ∈ X.

The rest of the proof is similar to the proof of Theorem 2.2.   ☐

Corollary 3.3. Let r > 1 and θ be nonnegative real numbers, and let f : X → Y be a mapping satisfying f(0) = 0 and

for all x, y, z ∈ X. Then there exists a unique additive mapping A : X → Y such that

for all x ∈ X.

Proof. Letting φ(x, y, z) := in Theorem 3.2, we get the desired result.   ☐

Theorem 3.4. Let φ : X3 → [0, ∞) be a function and let f : X → Y be a mapping satisfying f(0) = 0, (3.3) and

for all x, y, z ∈ X. Then there exists a unique additive mapping A : X → Y such that

for all x ∈ X.

Proof. It follows from (3.4) that

for all x ∈ X.

The rest of the proof is similar to the proofs of Theorems 2.2 and 2.4.   ☐

Corollary 3.5. Let r < 1 and θ be nonnegative real numbers, and let f : X → Y be a mapping satisfying f(0) = 0 and (3.5). Then there exists a unique additive mapping A : X → Y such that

for all x ∈ X.

Proof. Letting φ(x, y, z) := in Theorem 3.4, we get the desired result.   ☐